Topological pressure ($P(\varphi )={\lim }_n{n}^{-1}\log {\sum }_{|w|=n}{e}^{S_n\varphi (w\eta )}$)

For a finite full shift and a potential with summable variations, the displayed symbolic pressure exists and does not depend on the reference tail $\eta$. The almost-additive partition-function limit proves existence. The function $\gamma \mapsto P(\gamma U)$ is convex; finite-volume second derivatives are energy variances divided by block length. If $-B\le U\le -b<0$, its increments for $\delta >0$ lie between $-B\delta$ and $-b\delta$, so it is Lipschitz continuous, strictly decreasing, and has one positive zero.